# Convergence/Divergence of Integrals

## Main Question or Discussion Point

Determine whether the following integral converges or diverges without evaluating it.

$$\int_{0}^{1-}\frac{dx}{\sqrt{1-x^4}}$$

dextercioby
Homework Helper
That function behaves ~1/x^{2},so it should be integrable in that domain.U have to find a function which is greater than your function,however,it's integral on [0,1] needs to be finite...

$$\int_{0}^{1} \frac{dx}{\sqrt{1-x^{4}}} =\frac{1}{4}\left(\sqrt{\pi}\right)^{3}\frac{\sqrt{2}}{\Gamma^{2}\left(\frac{3}{4}\right)}$$

Daniel.

Hurkyl
Staff Emeritus
Gold Member
The problem point is x=1, not x=0. √(1 - x4) behaves like 2√(1-x) there.

P.S. 1/x2 isn't integrable over [0, 1]. :tongue2:

dextercioby
Homework Helper
I didn't say "integrable".And i didn't say =1/x^{2}...

Daniel.

mathwonk
Homework Helper
i believe it certainly converges. now let me think why i say that.

if y^2 = 1-x^4 then 2ydy = -4x^3 dx, so dx/sqrt(1-x^4) = dx/y = -2dy/4x^3. which makes perfectly good sense at x = 1.

huh???

formulas have always seemed like magic to me.

Last edited:
dextercioby
Homework Helper
Yeah,but it's still infinite,because your limits of integration will still involve 0...

Daniel.

saltydog
Homework Helper
dextercioby said:
That function behaves ~1/x^{2},so it should be integrable in that domain.U have to find a function which is greater than your function,however,it's integral on [0,1] needs to be finite...

$$\int_{0}^{1} \frac{dx}{\sqrt{1-x^{4}}} =\frac{1}{4}\left(\sqrt{\pi}\right)^{3}\frac{\sqrt{2}}{\Gamma^{2}\left(\frac{3}{4}\right)}$$

Daniel.
Can you explain how this integral is evaluated in terms of the Gamma function?

dextercioby
Homework Helper
I dunno.Basically,it returned my a value for the Legendre elliptic integral and then expressed this one in terms of Gamma factor...

Daniel.

dextercioby
Homework Helper
If u make a substitution,u can transform the integral to

$$\int_{1}^{+\infty} \frac{dx}{\sqrt{x^{4}-1}}$$

whose antiderivative,Mathematica finds

Daniel.

P.S.My Maple gave me the results...

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